A meshless p-PUFEM Fokker-Planck equation solver with automatic boundary condition enforcement

In this paper the Fokker-Planck equation is solved by the meshless particle partition of unity finite element method with automatic boundary condition enforcement. An improved cover generation algorithm is presented to automatically minimize interference between boundary nodes and interior nodes in the solution domain. Radial basis functions are used at boundary nodes to automatically damp the solution to a desired small value at boundaries, thus eliminating the need of penalty method for boundary condition enforcement. This achieves two important goals: (i) improves the conditioning of discretized reduced order system and (ii) eliminates spurious modes created by penalty method, thus improving the quality of point spectrum of the discretized Fokker-Planck operator. This is particularly important for systems without stationary solution, in which spurious modes can prevent the construction of accurate approximation. Results shown for a two dimensional nonlinear oscillator with stationary solution agree well with our former results while also decreasing the total number of spurious modes significantly. Further, an undamped simple pendulum is examined to demonstrate the applicability of this method for systems without stationary solution.